## New Problem: King of the Jungle Plays Tug of War

I have a couple of old problems I haven’t solved yet (1 2), but they’re kind of long, and this one is much shorter. The logic for this puzzle is from
Mathematical Puzzles, for Beginners and Enthusiasts by Geoffrey Mott Smith, where you can find it stated in a more boring form (problem 76). I’ve also modified it just a little.

The King of the Jungle is a lionhorse.

The lionhorse thinks it is totally amazing at tug of war. To prove it, he gets a strong rope, and assembles a bunch of mammals to play tug of war. He gets:

1. a mouse
2. an echidna
3. Mowgli the manchild
4. a tapir
5. an elephant
6. a beached whale

and prepares them to play tug of war. The mouse can pull with one mousepower (which is a unit of force). All the other mammals can exert a fixed force which can be measured in mousepowers. Assume everyone pulls with an integer number of mousepowers. To play a tug of war match, the lionhorse chooses two disjoint subsets of the assembled mammals as teams. They each tug on one side of the rope. The result is a win for the team with more total mousepower (it adds simply) or a tie if the mousepower is even.

The lionhorse doesn’t know his own mousepower (except that it’s also an integer), but he does know the mousepower of all the other animals he assembled. He plans to measure his own mousepower by participating in the matches.

Further, he picked the players with very careful consideration for their mousepower. In fact he’s set things up so he knows for sure that he’ll successfully measure his own mousepower as long as it’s under some specified ceiling. That is, he can measure a mousepower of $1, 2, 3, 4, ... n_{max}$ with $n_{max}$ some integer. But he’s also set things up to make $n_{max}$ as large as possible considering he has six other mammals around.

What is $n_{max}$?